On Explicit Holmes-Thompson Area Formula in Integral Geometry

TL;DR

This paper explores the Holmes-Thompson volume in Minkowski spaces, providing explicit integral formulas for area, and extends known results to higher dimensions using symplectic geometry and Crofton measures.

Contribution

It offers a detailed exposition of Holmes-Thompson volumes and derives explicit integral formulas for Minkowski plane areas, extending previous results to higher dimensions.

Findings

Explicit integral formula for Minkowski plane area.

Connection between symplectic structures and Holmes-Thompson volumes.

Extension of Alvarez's results to higher-dimensional Minkowski spaces.

Abstract

In this article, we give an exposition on the Holmes-Thompson theory developed by Alvarez. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the sphere-bundle. we show that it can be also obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the tangent bundle of the Minkowski unit sphere. We give detailed descriptions and expositions on Holmes-Thompson volumes in Minkowski space by the symplectic structure and the Crofton measures for them. For the Minkowski plane, a normed two dimensional space, we express the area explicitly in an integral geometry way, by putting a measure on the plane, which gives an extension of Alvarez's result for higher dimensional cases.